= Solution
For the zero-data <initial value problem>, choose a <causal Green function> satisfying $L_xG(x,\xi)=\delta(x-\xi)$ and $G=0$ for $x<\xi$. Continuity at $x=\xi$ prevents a derivative of the <Dirac delta distribution>, while integration across the source gives
$$
G(\xi^+,\xi)=0,\qquad\partial_xG(\xi^+,\xi)-\partial_xG(\xi^-,\xi)=1.
$$
For $x>\xi$, part (i) gives the shifted homogeneous solutions. The zero value at the join removes the cosine term, and the unit <derivative> jump fixes the sine coefficient. Therefore the <causal Green function of a damped oscillator> is
$$
\boxed{G(x,\xi)=\begin{cases}\dfrac1b e^{-a(x-\xi)}\sin[b(x-\xi)],&x\geq\xi,\\0,&x<\xi.\end{cases}}
$$
Equivalently, multiply the first expression by the <Heaviside step function> $H(x-\xi)$. The required solution is
$$
\boxed{y(x)=\int_0^{\infty}G(x,\xi)g(\xi)\,d\xi=\frac1b\int_0^xe^{-a(x-\xi)}\sin[b(x-\xi)]g(\xi)\,d\xi.}
$$
Its initial value and first <derivative> vanish: the interval has zero length at $x=0$ and the kernel vanishes at its upper limit. Differentiating twice shows that the upper-limit contribution to $y''$ is precisely $g(x)$, while the remaining homogeneous kernel terms cancel in $Ly$.
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